位移浅水波系统

(1+1)维位移浅水波系统(1DDSWWS)是结合流体力学和变分原理, 运用拉格朗日坐标而构造的浅水波方程. 综合流体在3个维度空间上的能量, 将1DDSWWS推广, 可推导出(2+1)维位移浅水波系统(2DDSWWS). 2DDSWWS的严格解可表示为椭圆函数积分, 这个椭圆函数积分可退化为雅可比椭圆周期函数解和孤立波解. 2DDSWWS的水面具有各种不同形态的孤子激发模式, 我们在2DDSWWS模型中也发现了孤子分子. 借用量纲分析的方法添加流体黏性项, 可以对理想的(2+1)维位移浅水波系统进行修正, 建立修正的2DDSWWS模型. 当黏性系数为零时, 修正模型将退化成理想模型. 修正的2DDSWWS模型的严格解可以很清晰地展示流体的黏性对流体运动的影响. 在连续性方程中保留高阶项, 重构拉格朗日函数, 可以得到全非线性(2+1)维位移浅水波系统(FN2DDSWWE). 在低阶近似下, 忽略某些高阶项, FN2DDSWWE可以退化成2DDSWWS模型.

Displacement shallow water wave system

One (1+1)-dimensional displacement shallow water wave system (1DDSWWS) is proposed to model the shallow water waves with Lagrange coordinate. Travelling wave solutions of the 2DDSWWS can be expressed as an elliptic function integral. Under some parameter constraints, the elliptic function integral can be reduced back to elliptic function solutions and solitary wave solutions. There are various soliton evolution patterns on the water surface of the 2DDSWWS. In particular, we succeed in identifying soliton molecules in the 2DDSWWS model. In this paper the displacement system is modified and the term related to the fluid viscosity is added to the model by means of dimensional analysis. For ideal fluids, the modified displacement system will degenerate to displacement system (M2DDSWWS). Some exact solutions of the M2DDSWWS clearly show the role of fluid viscosity. By preserving the higher-order terms in the continuity equation and reconstructing the Lagrangian function, one can obtain the fully nonlinear (2+1)-dimensional displacement shallow water wave equation (FN2DDSWWE). Making use of the low order approximation and neglecting the effect of some low order terms, the FN2DDSWWE can be reduced back to the 2DDSWWS.